Puzzle of the Week #128 - Five Cuboids in a Sphere

Fit five cuboids into a sphere of radius 1.

What is the greatest combined volume you can achieve?

Similarly to the Three Rectangles in a Circle puzzle a couple of weeks ago, you’ll find that a great deal of symmetry will help to maximize the volume. Again of course, the answer is not an integer, so I'll be happy to accept an answer to one decimal place.

Good luck!

Puzzle of the Week #127 - Odd One Out

Which of these words is the odd one out?


Notoriously, with odd-one-out questions, you will probably be able to find a reason why each of the words in turn should be the odd one. However, the true solution will be both convincing and satisfying, with even my choice of the odd word serving to support your hypothesis. Good luck!


Free Gift! More Binary Determination Puzzles

Here’s a little gift for you: an excel version of my Binary Determination puzzle.

Just stick it on your desktop and have a play if you ever get bored, or play it first thing in the morning to wake your brain up.

It will randomly generate one of a possible 100 billion billion billion puzzles (which is a lot more than the age of the universe in seconds). When you have completed a row or column the target number will turn green.

I’ve included macros to clear the board (but leave the target numbers as they are), or to begin a completely new game. You will need to enable macros and iterative calculation (circular references). I’ve protected the worksheet to prevent you from inadvertently changing cells that don’t need to be changed, but it’s not password protected so if you want to take a peek under the hood you are perfectly welcome!

You can download the .xlsm file from my Dropbox by following this link:



Puzzle of the Week #126 - Three Rectangles in a Circle

Fit three rectangles into a circle of radius 1.

What is the greatest combined area you can achieve?

3 rects in circle.JPG

(I usually like to have the solution be a whole number, but this is not possible in this case. It is however possible to express the solution in the form of √a + b, where a and b are whole numbers, but I’m also happy to accept an approximate answer to a couple of decimal places).

Bonus Puzzle of the Week #124a - Binary Determination

I’ve introduced a couple of refinements to my Binary Determination puzzle: firstly roughly equal amounts of ones and zeros (previously there were more ones at a ratio of about 2:1); secondly a more concise way of giving the solution

Place a 0 or 1 in each of the empty cells so that in each row and column a pair of 5-digit binary numbers can be read (therefore with decimal values between 0 and 31), such that the product of the two numbers in a particular row or column is shown at the end of that row or column.

Here is an example (using only 3-digit binary numbers), so for instance in the first column, 010 (2) multiplied by 011 (3) is equal to 6, (as there is a 6 at the foot of the first column) and similarly for all of the other columns and rows. Finally, read off the diagonal of each quarter of the grid and convert back into decimal to give the solution (3,3,2,3):

bindet eg.JPG

Here is the puzzle:


Puzzle of the Week #124 - Binary Determination

Here’s a brand new type of puzzle that I’ve just invented this week.

Place a 0 or 1 in each of the empty cells so that in each row and column a pair of 5-digit binary numbers can be read (therefore with decimal values between 0 and 31), such that the product of the two numbers in a particular row or column is shown at the end of that row or column.

Here is an example (using only 3-digit binary numbers), so for instance in the first column, 010 (2) multiplied by 011 (3) is equal to 6 (as that is the number at the foot of the first column), and similarly for all of the other columns and rows.

binary eg.JPG

Here is the puzzle:

binary puz1.JPG

Puzzle of the Week #123 - Forensic Probability

Imagine two games:

In one a person rolls a dice repeatedly and notes down the result. If a particular number appears three times, any subsequent roll of that number will be ignored. This is continued until ten numbers have been written down.

In another there is a pack of 18 cards, three each with the numbers 1 to 6 on them. 10 cards are drawn at random. The ten numbers are written down.

In both cases the result will be ten numbers between 1 and 6, with no number appearing more than three times. It will be impossible to know whether the ten numbers came from the dice or the cards.

Is it possible to detect, given the results of many repeated trials, which game someone is playing, or are the two games essentially identical? (I’m not for numbers, just general trends that one might look out for).

Paddocks is now out as a pdf ebook.

I know, I know, the printing and postage costs of print-on-demand are not inconsiderable, and whilst I believe Paddocks is better appreciated in its pocket size printed book form, I have now decided to make it available as an ebook.

Not only is it cheaper than the printed book (even with its 40% discount), but also you don't need to pay postage, and it is delivered immediately.

So what's stopping you accessing more than 100 great puzzles (mostly Paddocks, but some other interesting and novel varieties too)?


Puzzle of the Week #121 - parkrun PBs


Every Saturday I join around 200,000 other runners and take part in a free 5k run at one of 1000 parks around the world (see http://www.parkrun.com). It's easy to keep statistics on how my times are improving. Alongside here is a table of my twenty fastest parkrun times.

I recorded those times at 16 different parks, and revisited some of them. As well as recording the rank of each parkrun time overall, I also record what its rank was when I ran that time, so for instance when I ran at Long Eaton, it was the second fastest I had run, but because some subsequent runs have been faster, it is now only my 8th fastest time.

Using the information in the table, can you work out in what order I ran the twenty parkruns?

Puzzle of the Week #120 - Quality Control

A company orders in a variety of pieces of metal of lengths:

7cm        17cm     18cm     19cm     25cm     37cm     44cm

46cm     63cm     65cm     82cm     83cm     90cm     100cm

The quality controller, whose task it is to check the length of the incoming parts is correct, has figured out that he can use a straight rod of a particular length, and mark a series of points on it so that all of the lengths in the above list can be measured, either between two of the marks, or from an end to one of the marks. He uses the minimum possible number of marks.

How long is the rod, how many marks were made, and where were they positioned?

Puzzle of the Week #118 - Paddocks

To celebrate the release of the Paddocks book: (click here), here is a brand new Paddocks puzzle you won’t even find in the book.

paddocks new.JPG

Draw fences between some of the posts so that each post is at the junction of exactly THREE fences.

These fences will divide the field into several PADDOCKS; any paddock whose area is greater than a single triangle will contain a NUMBER, which will indicate the area of the paddock, or in other words the number of TRIANGLES that make up the paddock.

The boundary fence is already in place, so any post on the boundary only needs one more fence emerging from it in order to make up its full complement.

For example:

PADDOCKS example.jpg

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Paddocks: The Book

I’m proud to announce the publication of my third book: Paddocks! 


I’m very proud of this book, it’s small but perfectly formed. Perfect for keeping it you pocket and taking out on long train journeys. Packed with over 100 hand-crafted puzzles, mostly Paddocks, but a smattering of other great puzzles to provide little breaks throughout the book. 

To let you know the background, I came up Paddocks a few years ago, and it was very popular with the readers of my magazine, and later my website. It gained more traction when some of them were featured in Alex Bellos’ column in The Guardian, where it was very well received. Like Sudoku, they can be solved by pure deduction, and there is only one possible solution. Also like Sudoku, one deduction here might have a knock-on effect over there, as you gradually move towards the answer. 

The task is to add fences within a field to divide it into Paddocks according to some very simple rules. It’s easy to learn, but difficult to master (the final puzzle in the book took me hours to test-solve!). 

As it's print-on-demand the list price is quite high, but if you are a fan of Paddocks, or of logic puzzles in general, it’s well worth it. Amazon and other retailers take a big share, so the good news is if you go straight to lulu.com via the link above you get a massive 40% discount on the list price. 

Thanks for your support! 

Elliott Line 




Puzzle of the Week #117 - Bike Ride

I recently went on a bike ride where I noticed curiously that the start point, the finish point and the rest point were all exactly the same distance from my house.

The first leg of the ride involved riding 23 miles due east and then 7 miles due north, whereupon I arrived at the rest point.

The second leg of the ride involved riding 15 miles due east and then 9 miles due north to the finish.

How far is my house from the start/rest/finish points?

Puzzle of the Week #114 - Four Spheres in a Box

If you place a sphere inside a cube-shaped box that is only just big enough to contain the cube, the sphere will 52% fill the cube. The closest fraction to this, with a single digit numerator and denominator, is 1/2.

If you were to place 4 identical spheres inside a cube shaped box only just big enough to contain them, what proportion of the box filled by the spheres (to the closest single digit fraction)?